Muon g-2 Hadronic Vacuum Polarization

Transcription

1 Muon g-2 Hadronic Vacuum Polarization Christoph Lehner (BNL) RBC and UKQCD Collaborations May 18, 216 TUM

2 The anomalous magnetic moment Potential of particle in magnetic field with V (x) = µ B(x) (1) µ = g where S is the spin of the particle. ( e ) S, (2) 2m Relativistic description with classical photon (Dirac) yields g = 2 (3) but taking into account QFT yields non-zero anomalous magnetic moment a = (g 2)/2. (4) 1 / 35

3 The anomalous magnetic moment These anomalous moments are measured very precisely. For the electron (Hanneke, Fogwell, Gabrielse 28) a e = (28) (5) yielding the currently most precise determination of the fine structure constant α = 1/ (33) (6) via a 5-loop QED computation (Aoyama, Hayakawa, Kinoshita, Nio 215). 2 / 35

4 The anomalous magnetic moment a requires QFT: a can be expressed in terms of scattering of particle off a classical photon background For external photon index µ with momentum q the scattering amplitude can be generally written as ( ie) [γ µ F 1 (q 2 ) + iσµν q ν ] 2m F 2(q 2 ) with F 2 () = a. (7) 3 / 35

5 The muon anomalous magnetic moment The muon anomalous magnetic moment promises to be useful to discover new physics beyond the standard model (SM) of particle physics. In general, new physics contributions to a l are given by a l al SM (ml 2/Λ2 NP ) for lepton l = e, µ, τ and new physics scale Λ NP. With l = τ being experimentally inaccessible, l = µ promises good sensitivity to new physics. Example contributions: one-loop MSSM neutralino/smuon and chargino/sneutrino contributions to a µ 4 / 35

6 The muon anomalous magnetic moment Currently a tension of more than 3σ exists: Total SM prediction (PDG) (4.9) BNL E821 result (6.3) a EXP µ a SM µ = (27.6 ± 8.) 1 1 (8) 5 / 35

7 And new experiments (Fermilab E989 and J-PARC) promise a 4 reduction in experimental uncertainty: 6 / 35

8 And new experiments (Fermilab E989 and J-PARC) promise a 4 reduction in experimental uncertainty: Final Report E821, 26 quency. In the presence of both E and B fields, and in the case that β o both E and B,theexpressionfortheanomalousprecessionfrequency ω a = q ( [a µ B a µ 1 ) ] β E. (5) m γ 2 1 c 6 / 35

9 Hadronic contributions to a µ Contribution Value 1 1 Uncertainty 1 1 QED (5 loops) EW HVP LO HVP NLO HVP NNLO Hadronic light-by-light Total SM prediction BNL E821 result Fermilab E989 target 1.6 A reduction of uncertainty for HVP and HLbL is needed. For HLbL only model estimations exist. First-principles non-perturbative determination desired. 7 / 35

10 Classification of hadronic contributions: HVP LO HVP NLO HVP NNLO HLbL / 35

11 The dispersive approach to HVP LO

12 e common feature of all hadronic contributions discussed in this paper will t theythe aredispersion self-energy relation insertions in the photon propagator. We introduce t arization function Π µν for the hadronic insertion with Π µν (q) = i ( q µ q ν g µν q 2) Π(q 2 ) Π(q 2 ) = q2 π 4m 2 π ds ImΠ(s) s q 2 s. e substitution rule from a photon propagator to a renormalized propagator w ronicallows insertion for follows: the determination of aµ HVP from experimental data via i g µν q 1 ds ImΠ(s) ( ) g µν i. 2 [ π 4m 2 π s q 2 s ( HVP LO αmµ ) 2 E 2 The calculation aµ = of higher-order kernel ds Rexp γ (s) ˆK(s) 3π functions reduces to thecomputation 4m -loop QED diagrams where one 2 s 2 + ds RpQCD γ (s) ˆK(s) ] photon is massless and the other photon has t ctive mass π E 2 s 2, s. The s integration, which has been shifted to the very end, R γ (s) = σ () (e + e γ hadrons)/ 4πα2 formed over experimental data using 3s R(s) = σ(e+ e hadrons) Experimentally with or without additional hard =12πphoton ImΠ(s) (ISR: e + e γ ( hadrons)γ) σ (e + e µ + µ ) + + 4πα 2 9 / 35

13 BESIII 215 update: KLOE ±.4 ± 2.3 ± 2.2 BaBar ± 2. ± 1.9 KLOE ±.9 ± 2.3 ± 2.2 KLOE ± 1.2 ± 2.4 ±.8 BESIII ± 2.5 ± ππ,lo -1 a µ (6-9 MeV) [1 ] Figure 7: Our calculation of the leading-order (LO) hadronic vacuum polarization 2 contributions to (g 2) µ in the energy range 6-9 MeV from BESIII and based on the data from KLOE 8 [6], 1 [7], 12 [8], and BaBar [1], with the statistical and systematic errors. The statistical and systematic errors are added quadratically. The band shows the 1 range of the BESIII result. [18] S. Jadach, B. F. L. Ward and Z. Was, Comput. Phys. Quantum Field Theory, Vol.2,USA,Addison-Wesley, 1 / 35

14 Jegerlehner FCCP215 summary: final state range (GeV) a had(1) µ 1 1 (stat) (syst) [tot] rel abs (.28, 1.5) (.39) ( 2.68)[ 2.71].5% 39.9%! (.42,.81) (.42) (.95)[ 1.4] 3.% 5.9% ( 1., 1.4) (.48) (.79)[.92] 2.7% 4.7% J/ 8.94 (.42) (.41)[.59] 6.6% 1.9%.11 (.) (.1)[.1] 6.8%.% had ( 1.5, 2.) 6.45 (.21) ( 2.8)[ 2.8] 4.6% 42.9% had ( 2., 3.1) (.12) (.92)[.93] 4.3% 4.7% had ( 3.1, 3.6) 3.77 (.3) (.1)[.1] 2.8%.1% had ( 3.6, 9.46) (.4) (.1)[.4].3%.% had ( 9.46,13.) 1.28 (.1) (.7)[.7] 5.4%.% pqcd (13.,1) 1.53 (.) (.)[.].%.% data (.28,13.) (.89) ( 4.19)[ 4.28].6%.% total (.89) ( 4.19)[ 4.28].6% 1.% Results for a had(1) µ 1 1. Update August 215, incl SCAN[NSK]+ISR[KLOE1,KLOE12,BaBar, BESIII ] F. Jegerlehner FCCP 215, Capri, Sept. 1-12, / 35

15 Jegerlehner FCCP215 summary (τ e + e ): excl. NSK (e + e ) ± 6.9 NSK+KLOE (e + e ) ± 6.6 NSK+BaBar (e + e ) ± 6.3 NSK+BESIII (e + e ) ± 6.8 ALL (e + e ) ± 6.2 incl. NSK (e + e + ) ± 5.9 NSK+KLOE (e + e + ) ± 5.6 NSK+BaBar (e + e + ) 182. ± 5.4 NSK+BESIII (e + e + ) ± 5.8 ALL (e + e + ) ± 5.3 experiment BNL-E821 (world average) 28.9 ± 6.3 best [3.3 ] [3.9 ] [3.1 ] [3.4 ] [3.5 ] [3.6 ] [4.1 ] [3.3 ] [3.7 ] 3.8 [3.8 ] aµ F. Jegerlehner FCCP 215, Capri, Sept. 1-12, / 35

16 Lattice QCD

17 Overview of first-principles lattice QCD results a µ R-ratio (PDG) RBC212 ETMC213 HPQCD216 HPQCD216(CON) On-going efforts by ETMC, HPQCD+MILC, Mainz, RBC+UKQCD,... HPQCD216(CON) neglects the systematic error estimates for the HVP disconnected and QED/isospin-breaking corrections. 13 / 35

18 The Leading Order Hadronic Vacuum Polarization Quark-connected piece with > 9% of the contribution with by far dominant part from up and down quark loops (Below focus on light contribution only) Quark-disconnected piece with 1.5% of the contribution (1/5 suppression already through charge factors); arxiv: , accepted for PRL esti- QED and isospin-breaking corrections, mated at the few-per-cent level 14 / 35

19 HVP quark-connected contribution Biggest challenge to direct calculation at physical point is to control statistics and potentially large finite-volume errors (Estimated at O(1%) Aubin et al. 215) Finite-volume errors are exponentially suppressed in the simulation volume but seem to be sizeable in QCD boxes with m π L = 4 Statistics: for strange and charm solved issue, for up and down quarks existing methodology (such as HPQCD moments approach) less effective 15 / 35

20 HVP quark-connected contribution Starting from e iqx J µ (x)j ν () = (δ µν q 2 q µ q ν )Π(q 2 ) (9) x with vector current J µ (x) = i f Q f Ψ f (x)γ µ Ψ f (x) and using the subtraction prescription of Bernecker-Meyer 211 Π(q 2 ) Π(q 2 = ) = ( cos(qt) 1 q ) 2 t2 C(t) (1) t with C(t) = 1 3 x j=,1,2 J j( x, t)j j () we may write a HVP µ = w t C(t), (11) t= where w t captures the QED part of the diagram. 16 / 35

21 Remark: the HPQCD moments method is also nicely described in this position-space representation a HVP µ = w t C(t). (12) t= Expanding w t in powers of t reveals the leading contribution at t 4 and matches to the moments method. The Bernecker-Meyer kernel has the same statistical and finite-volume benefits as the HPQCD moments method. 17 / 35

22 Integrand w T C(T ) for the light-quark connected contribution: 1 8 N conf = 82 Lattice temporal integrand NLO FV ChPT temporal integrand 6 4 a µ T m π = 14 MeV, a =.11 fm (RBC/UKQCD 48 3 ensemble) Statistical noise from long-distance region 18 / 35

23 Approaches to the long-distance noise problem: HPQCD 216: only uses lattice data up to.5fm 1.5fm, beyond that multi-exponentials from fit RBC in progress: improved stochastic estimator Z 2 sources/config Multi-step AMA with 2-mode LMA (same cost) 1 48 Z 2 sources/config Multi-step AMA with 2-mode LMA (same cost) a µ a µ T T 19 / 35

24 Low-mode saturation for physical pion mass (here 2 modes): N conf = 82 Full temporal integrand Sloppy temporal integrand No new physics Low-mode only integrand NLO FV ChPT temporal integrand 4 a µ T 2 / 35

25 Another approach to control statistics: It is potentially helpful to define stochastic estimator for strict upper and lower bounds of a µ which has reduced statistical fluctuations C.L. et al Strict upper bound Strict lower bound 8 6 a µ T up and down loop shown here: data shown here is from early stages of computation with 5% statistical error, currently at around 2% statistical error. Within the next year our current setup can produce a continuum limit with 1% statistical error. 21 / 35

26 A closer look at the NLO FV ChPT prediction (1-loop sqed): We show the partial sum T t= w tc(t) for different geometries and volumes: 8 7 a conn µ 11 (NLO FV ChPT) L T for 96 x 48 3 (short time) 2 L T for 128 x 64 3 (short time) L T for 192 x 96 3 (short time) 1 L T for 48 3 x 96 (large time) L T for 64 3 x 128 (large time) L T for 96 3 x 192 (large time) T 22 / 35

27 From Aubin et al. 215 (arxiv: v2) FIG. 4: Comparison of A1 (ˆq2 ) A1 (ˆq2 ) between MILC asqtad lattice data (blue FIG. points) 5: Comparison and of A1 (ˆq2 ) A 44 ) between MILC asqtad lattice data (blue points) and 1 lowest-order SChPT (red points). lowest-order SChPT (red points). MILC lattice data with m πl = 4.2, m π 22 MeV; Plot difference of Π(q 2 ) from different irreps of 9-degree the continuum limit. However, the slopes are also vastly di erent, and this We is amay physical also consider di erences between di erent representations, which also probes the e ect, already rotation observed symmetry in Ref. [12]. of spatial The slope components of the vacuumversus polarization NLO sizefv at of low finite-volume ChPT q 2 is prediction e ects. In(red Fig. dots) 5 we show the di erence A1(ˆq 2 ) A 44 (ˆq2 ), for the dominated by the resonance, but this resonance (and others) are absent in Eq. (2.12) lattice data, and computed in SChPT. To extract A 44 (ˆq2 )from µ (ˆq) weneedatleastone 1 DespiteWhile these di erences, thethere absolute are useful lessonsvalue to be learned ofrom a Fig. 3. The subtracted value A1(ˆq 2 ) is an order of magnitude closer to the infinite-volume µ spatial is component poorly of thedescribed momentum to not vanish, byimplying thethat two-pion ˆq /L 2 =.18 GeV 2 for points these than points. the All observations made above about the di erence A1(ˆq 2 ) A1(ˆq 2 )apply unsubtracted value, A1(ˆq 2 ). Clearly, lesson is that one should carry out herethe as subtraction (2.6) contribution, (at least for the A1 representation). the volume This was alreadydependence observedwe empirically note the di erence in may of scale beon the described vertical axis between sufficiently Figs. 4 and 5, consistent with well, with the di erence between lattice data and ChPT now averaging about 1. Ref. [22], and see here that this observation is theoretically supported the by ChPT. fact that Furthermore, see that A1(ˆq 2 1 both A1(ˆq 2 )and A 44 (ˆq2 )aremuchclosertotheinfinite-volumelimitthan well to use )and ChPT A 44 (ˆq2 )straddletheinfinite-volumeresult,suggesting to control FV A1(ˆq errors 2 ). We find that athethe pattern 1% is very similar level; for other this representations. needs 1 that also in lattice QCD the true value of (q 2 )liesinbetweenthesetwo. 17 Of course, further one would like scrutiny to test whether these lessons from lowest-order SChPT also B. E ects on a apply to the actual lattice data. While no lattice data are available in infinite volume, HVP µ it is possible to compare finite-volume di erences predicted by SChPT to such di erences computedaubin from the lattice et data. al. In Fig. find 4 we show anthe O(1%) di erence A1(ˆq 2 ) finite-volume Finally, A1(ˆq 2 while it is already )inthe error clear thatfor there are msignificant finite-volume e ects in the low-ˆq 2 vacuum polarization, we consider the question of how π L = 4.2 they propagate to the anomalous region, both on the lattice and computed in lowest-order SChPT. This di erence is a magnetic moment itself. We will, in fact, compare the quantity a LO,HVP µ [ˆq 2 pure finite-volume e ect. Clearly, SChPT does a very good job of describing the lattice data, max] with the based on the A choice ˆq max with all red points within less than 1 of the 1 A 44 blue points. This is remarkable, especially 2 =.1 GeV in 2, in order to be certain that di erences are due to finite volume, and not to lattice spacing e ects. view of the fact that lowest-order SChPT does such a poor job of describing the full lattice 18 data for (ˆq 2 We fit the data for A1 and ), as we noted above. A 44 with a [, 1] Padé [19], or a quadratic conformally 1 1 difference (right-hand plot) 23 / 35

28 Compare difference of integrand of (spatial) and (temporal) geometries with NLO FV ChPT (A 1 A 44 1 ): 2 15 N conf = 82 Full spatial-temporal integrand NLO FV ChPT spatial-temporal integrand NLO FV ChPT + ρ back prop 1 a µ T m π = 14 MeV, a =.11 fm (RBC/UKQCD 48 3 ensemble) 24 / 35

29 It may be worth verifying that the O(1%) finite-volume error estimate from Aubin et al. was not spoiled by a backwards-propagating ρ: N conf = 82 Full spatial integrand Sloppy spatial integrand No new physics Low-mode only integrand NLO FV ChPT spatial integrand 4 a µ T 25 / 35

30 HVP quark-disconnected contribution First results at physical pion mass with a statistical signal RBC/UKQCD arxiv: , accepted by PRL Statistics is clearly the bottleneck New stochastic estimator allowed us to get result HVP (LO) DISC aµ = 9.6(3.3) stat (2.3) sys 1 1 (13) from 2 configurations at physical pion mass and 45 propagators/configuration. 26 / 35

31 Our setup (arxiv: ): C(t) = 1 3V j=,1,2 t V j (t + t )V j (t ) SU(3) (14) where V stands for the four-dimensional lattice volume, V µ = (1/3)(V u/d µ V s µ), and V f µ(t) = x Im Tr[D 1 x,t; x,t (m f )γ µ ]. (15) We separate 2 low modes (up to around m s ) from light quark propagator as D 1 = n v n (w n ) + D 1 high and estimate the high mode stochastically and the low modes as a full volume average Foley 25. We use a sparse grid for the high modes similar to Li 21 which has support only for points x µ with (x µ x µ () ) mod N = ; here we additionally use a random grid offset x µ () per sample allowing us to stochastically project to momenta. 27 / 35

32 Combination of both ideas is crucial for noise reduction at physical pion mass! Fluctuation of V µ (σ): σ σ / (96 / N) Light Light - Strange Light Highmode - Strange Sparse grid spacing N idea of O(1/M ) used in Ref. [12 gree, and theref in the lower the powerful can strange quark co ening strategy. for the case at numerical discu We use 45 sto measure on 21 M of the tice cuto a 1 UKQCD collab modes we find t certainty for a HV µ FIG. 2. Noise of single vector operator loop as a function was employed t Since C(t) is the autocorrelator of V µ, we can create a stochastic estimator whose noise is potentially reduced of sparse grid spacing N. The figure at the bottom normalizes the noise by taking into account the additional volume tion presented in ple sources on linearly in the number of random samples, hence the normalization in the lower panel averaging for smaller values of N. 28 / 35

33 Result for partial sum L T = T t= w tc(t): a DISC µ L T=2 Partial contribution of lattice data for t T T T FIG vol com tec one QC cis exp pre per str For t 15 C(t) is consistent with zero but the stochastic noise is FIG. 5. The sum of L T and F T defined in Eqs. (13) and (14) t-independent and w t t 4 such that it is difficult HVP (LO) to identify DISC a has a plateau from which we read o aµ. The plateau region based only on this plot lower panel compares the partial sums L T for all values of HVP (LO) DISC 29 / 35 W

34 Resulting correlators and fit of C(t) + C s (t) to c ρ e Eρt + c φ e E φt in the region t [t min,..., 17] with-1e-5 fixed energies E ρ = 77 MeV t and E φ = 12. C s (t) is the strange connected correlator. 4e-5 p =.12, c ρ = -.17(9), c Φ =.16(5) C(t) + C s (t) C(t) 1e-5 F T (r) where c r range r an T, L T is e of T from sum L T + FIG. 3. Zero-momentum projected correlator C(t) andc(t)+ the exten Cs(t). A correlated fit of (77) and (12) exponentials via Based c e E t +c e E t in the region t 2 [11,...,17] to C(t)+Cs(t) ferred fitr = [12, yields a p-value of.12. We use fixed energies E =77MeV 4 and E = 12 MeV and fit parameters c and c. tent resul.25 We now define the partial sums for L T a c ρ, p>.5 tion of ou 3e-5 c.2 Φ, p>.5 TX dependen L T = w tc(t), (13).15 L T =2 an 2e-5 t= mating a.1 tmax X 1e-5 F T (r) = w t(c r e E t + c r e E t the value C s(t)), (14).5 a plateau t=t +1 suppress where c r and c r HVP (LO are the parameters of the fit with fitrange -.5r and t max = 24 for our setup. For su ciently large We exp aµ -1e T, L T is expected to exhibit a plateau region as function tion volum t t min HVP (LO) DISC to domin of T from which we can determine aµ.the With resp sum L T +F T is also expected to exhibit such a plateau to FIG. 3. Zero-momentum projected correlator C(t) andc(t)+ FIG. the4. extent Coe cients that the andmodel p-values inof Fa T fit describes of c e E t the +cdata e E well. t proxy for Cs(t). A correlated fit of (77) and (12) exponentials via to C(t)+Cs(t) in the region t 2 [tmin,...,17]. connected Based on Fig. 4, we choose r = [11,...,17] as preferred a spectral fit-range representation. result at c ewe E t fit +ctoe C(t) E t in + the C s (t) region insteadt 2 [11,...,17] of C(t) since to C(t)+Cs(t) the former has to determine F T but a cross-check with yields a p-value of.12. We use fixed energies E =77MeV the contin r = [12,...,17] has been performed yielding consistent result. Figure 5 shows the resulting plateau-region and E = 12 MeV and fit parameters c and c. above Wick contractions, we can represent it as a sum over individual exponentials C(t)+C s(t) = P tent with Emt We could use this model alone for the m cme.25 for L T long-distance and L T + F T tail to help ate for th. In order to avoid contamination of our first-principles computation with the model- with c c ρ, p>.5 m 2 R and E m 2 R +. The coe cients c m can combined identify a c.2 Φ, p>.5 plateau but it would miss be negative the because two-pion positivity tail arguments HVP only (LO) apply DISC to tially mis some dependence individual ofwick F T, we contractions determinein aµ Eq. (12) but not from finite-volu.15 necessarily L T =2 and toinclude the sum. F T =2 as systematic uncertainty estimating a potentially missing long-time tail. We choose 3 computat / (ChPT) v 35

35 We therefore additionally calculate the two-pion tail for the disconnected diagram in ChPT: 5 a DISC µ 11 (ChPT) L T for 32 3 x 64 lattice L T for 48 3 x 96 lattice L T for 64 3 x 128 lattice L T for 96 3 x 192 lattice T 24 FIG. 6. The leading-order pion-loop contribution in finitevolume ChPT as function of volume. 31 / 35

36 a -15 We then pick a point in the potential plateau region such as T = 2 and use -2 a combined estimate of the resonance model and the two-pion tail to estimate -25 t=t +1 w tc(t) as a systematic uncertainty. a DISC µ L T=2 Partial contribution of lattice data for t T T T a -7-8 FIG. 6. Th volume ChP computatio technique e one of the QCD compu cision, nece experimenta presented h perimental straightforw Combined with an estimate FIG. 5. The sum of L T of discretization and F T errors, we find defined in Eqs. (13) and (14) HVP (LO) DISC hashvp a plateau (LO) DISC from which we read o aµ. The a lower µ = 9.6(3.3) panel compares the partial sums stat (2.3) L T sys 1 1. (16) for all values of We would HVP (LO) DISC T with our final result for aµ with its statistical rators for he 32 / 35

37 HVP QED corrections Largely unexplored but finite-volume errors are likely substantial New methods with potential to control large finite-volume errors in lattice QCD+QED simulations may prove useful (C boundary conditions Lucini et al. 215, massive QED Endres et al. 215, QED C.L. et al. Lattice 215) We are actively working on this measurement using technology similar to our on-going hadronic light-by-light calculation; we also have first results in quenched QED at heavier pion mass. 33 / 35

38 Lattice status and prospects: First-principles determination of the HVP contribution comparable with Fermilab E989 uncertainty (.3% uncertainty on HVP) is very challenging Substantial progress by various groups in the last year both for the HVP light connected and disconnected contributions Active effort on necessary sub-leading contributions such as QED/isospin-breaking corrections 34 / 35

39 Thank you FERMILAB-FN-992-E 35 / 35

40 Experimental setup: muon storage ring with tuned momentum of muons to cancel leading coupling to electric field spin precession frequency. In the presence of both E and B fields, and in the case that β is perpendicular to both E and B,theexpressionfortheanomalousprecessionfrequency becomes ω a = q ( [a µ B a µ 1 ) ] β E. (5) m γ 2 1 c The coefficient of the β E term vanishes at the magic momentum of 3.94 GeV/c, where Because of parity violation in weak decay of muon, a correlation between muon spin and decay electron direction exists, which can be used to measure the anomalous precession frequency ω a : γ =29.3. Thus a µ can be determined by a precision measurement of ω a and B. At this magic momentum, the electric field is used only for muon storage and the magnetic field alone determines the precession frequency. The finite spreadinbeammomentumandvertical betatron oscillations introduce small (sub ppm) corrections to the precession frequency. The longitudinally polarized muons, which are injected into the storage ring at the magic momentum, have a time-dilated muon lifetime of 64.4 µs. A measurement period of typically 7 µs followseachinjectionor fill. Thenetspinprecessiondepends on the integrated field seen by a muon along its trajectory. The magnetic field used in Eq. 5 refers to an average over muon trajectories during the course of the experiment. The trajectories of the muons must be weighted with the magnetic field distribution. To minimize the precision with which the average particle trajectories must be known, the field should be made as uniform as possible. FIG. 2: Distribution of electron counts versus time for the 3.6 billion muon decays in the R1 µ data-taking period. The data is wrapped around modulo 1 µs. Because of parity violation in the weak decay of the muon, a correlation existsbetween representative electron decay time histogram is shown in Fig. 2.